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..Macroeconomics questions

1.

Section 3. Present value and recursive asset pricing formula (i) Annuity pricing a) An annuity is a nancial instrument that pays out a constant amount of income per period for as long as the annuitant [i.e. annuity purchaser) is alive. Let P be the expected present value of annuity payments. Let n denote annuity income payment per period, to. denote the probability of dying by next period {aka mortality rate) and 1' denote the interest rate. Use the recursive asset pricing formula _u+lii-"'t _ 1+r to show that a. r+m' b} Suppose that an elderly annuitant has 500,000 in wealth and uses this wealth to purchase an annuity. Calculate the annual payment that the annuitant should expect if r = 0.05 and m = 0.03 per year. Explain how your answer is related to equilibrium economic prot of a nancial institution that issued the annuity. P: (ii) Life insurance premium A life insurance policy is a nancial contract that collects a premium of p per period from the insured and pays out a lump sum benet 3 upon the death of the insured. Let n: denote the mortality rate, r denote the interest rate and V denote the expected present value of economic prots of the insurance company on this policy. Use the recursive asset pricing formula p+EV 1+r to express equilibrium life insurance premium, 3:, through exogenous variables. Explain your answer and show all work. (iii) Loan caimdarors Suppose that loan of size S is repaid over T periods with equal per-period payments in and interest rate on the loan 1". Initial loan size has to be equal to the present value of the payments: P P P 1+r+(1+r)2+'"+(1+r)7 a) Calculate a monthly payment on the following mortgage. Annual interest rate is r = 0.06, mortgage term is 30 years, loan size S = 200,000. (Hint: since you are asked about the monthly payment, present value should be discounted at a monthly frequency. Start with guring out the monthly interest rate and the mortgage term expressed in months, then apply the formula). b) Jill would like to buy a car that costs 5 = 20,000. The dealer o'ers her a 60-month loan with 0 down and a monthly payment 19 = 421. Calculate the annual interest rate on this loan using Excel. Take the following steps: 1. Generate a column of 40 monthly interest rates, withr = [0.001,0.002. .0040}, say column A, starting with Al . 2. Use the geometric series formula to express the present value of payments through p, T and rm, where rm is the monthly interest rate. Plug in the numeric value for p, T and put this formula into cell Bl. Have the formula refer to the interest rate from cell A1. Select 3103100 and press CTRL+D. This will copy the formula and generate a column of present values that correspond to di'erent monthly interest rate. Pick an interest rate from a row whose cell B value corresponds most closely to S. 3. Convert the monthly interest rate rm that you found on step 2 into annual using the compounding formula 1 + r\" = (1 + rm)\". y: S: Problem Consider a model where money enters the utility function because of the transaction services it provides to households. The economy is populated by a continuum of identical infinitely-lived households. The representative households wants to maximize his lifetime utility: U = E. > gu ( C+, It, P. M+ 1=0 where 6 6 (0, 1) is the discount factor, C, is consumption, L, is labor, Me is nominal holdings of money (cash), and Pe is the price level in the economy. Foor simplicity, assume utility takes the following form: Ll+x 2 (Ct , Lt ; P. = In C+ + yln p. where y > 0, x >0,7>0 1+ x Notice that y indexes the importance of real money holdings for transactions, while y indexes the disutility from working. Utility is then increasing in consumption C, and real cash-, but decreasing in labor Ly. The household's budget constraint in every period t is: P.C+ + M+ + B+ = R+-1B+-1 + Me-1+ W.L+ + It where B, is a government bond which delivers a riskless gross return R, (hence Rt-1B,-1 are payments in t from a bond purchased in t - 1, including both the principal and the interests). The term It stands for transfers received by the central bank. As in class, we are going to denote gross inflation in period t as The firm's problem is the following. It chooses labor L, to maximize profits At = PYt - WL, subject to It = A.Ly, where a e (0, 1) under flexible prices and perfect competition. Answer to the followings.now on, let's suppose that At = 1 at all times, so that there is no more uncertainty in the model. This implies that you can drop the expectation operator from the Euler equation 1. Consider two alternative monetary policies: - the central bank controls nominal money growth: M. = AM_,, for & 2 1 (Money Growth Rule, MGR) - the central bank directly controls the interest rate R (Interest Rate Rule, IRR) Under the MGR, as seen in class, we have that my = mi_, -it Under the IRR, assume that Re = 4, it, where with the coefficient ( >1. Answer to the followings. a) Suppose the central banks adopts the MGR. Find the equilibrium inflation rate n, of the model. HINT: this is basically what we have done in class. First you have to solve for the equilibrium value of my, then recall that me = In , and then use the definition of my= . b) Now, suppose the central banks sets the interest rate R (entering the Euler equation (1)) according to Re =_*+. In this case, we assume that the steady state inflation is a > 1. You can think of this also as a choice of the central bank (that is what the Fed actually does). Find the equilibrium inflation rate x, and the equilibrium growth rate of nominal money, namely M.-1 M. HINT: as for a), you will have to solve a difference equation. However, this time the difference equation will be in terms of a, not m. You have to find the unique stable solution of such difference equation.4. Find the equilibrium expressions for output It, consumption C+, labor L, and the real wage We = p They should be (non-linear) functions only of At- 5. After linearizing the model around its steady state, you can find the following expressions for the (lin- earized) Euler equation and money demand: Of = E.C+1 - (R+ - Exit+1 ) (1) mi = noct -nRR+ (2) where no, 78 > 0 (they are composite coefficients depending on model parameters), and, as in class, a "hat" denotes the percentage deviation of the variable from its steady state. Ex: C, = In . FromA researcher claims that the best model to explain the relationship between spending and teacher's salary is a linear model, as follows: Where Salary is the average teacher's salary by state Spending is the spending per student in each state a. Run this model and calculate the following: SUMMARY OUTPUT Regression Statistics R Square Standard Error Observations ANOVA df SS MS F Significance F Regression Residual Total Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept Spending b. Explain the meaning of the coefficients for this regression c. If a State increased Spending per student by $2500, how much would the expected Salary will increase? d. Calculate the predicted line of your model. Draw a graph with the variables Salary and spending and include your estimated line in your graph e. Check the required conditions of your model. Show graphs and explain your results. f. Test if the coefficient for spending is equal to 3. Show your results and explain g. Using SSE and the R', assess your model. h. If a given state has a Spending per student of 14,000, what is the mean salary for teachers? What is the predicted salary for that specific State? (Calculate 95% confidence intervals for each case).AVERAGE SALARY OF PUBLIC SCHOOL TEACHERS BY STATE, 2005-2006 Salary Spending Connecticut 60,822 12,436 Illinois 58,246 9,275 Indiana 47,831 8,935 Iowa 43,130 7,807 Kansas 43,334 8,373 Maine 41,596 11,285 Massachusetts 58,624 12,596 Michigan 54,895 9,880 Minnesota 49,634 9,675 Missouri 41,839 7,840 Nebraska 42,044 7,900 New Hampshire 46,527 10,206 New Jersey 59,920 13,781 New York 58,537 13,551 North Dakota 38,822 7,807 Ohio 51,937 10,034 Pennsylvania 54,970 10,711 Rhode Island 55,956 11,089 South Dakota 35,378 7,911 Vermont 48,370 12,475 Wisconsin 47,901 9,965 Alabama 43,389 7,706 Arkansas 44,245 8,402 Delaware 54,680 12,036 District of Columbia 59,000 15,508 Florida 45,308 7,762 Georgia 49,905 8,534 Kentucky 43,646 8,300 Louisiana 42,816 8,519 Maryland 56,927 9,771 Mississippi 40,182 7,215 North Carolina 46,410 7,675 Oklahoma 42,379 6,944 South Carolina 44,133 8,377 Tennessee 43,816 6,979 Texas 44.897 7,547 Virginia 44,727 9,275 West Virginia 40,531 9,886 Alaska 54,658 10,171 Arizona 45,941 5,585 California 63,640 8,486 Colorado 45,833 8,861 Hawaii 51,922 9,879 Idaho 42,798 7,042 Montana 41,225 8,361 Nevada 45,342 6,755 New Mexico 42,780 8,622 Oregon 50,911 8,649 Utah 40,566 5,347 Washington 47,882 7,958 Wyoming 50,692 11,596